Someone recently pointed out to me that most English-speakers will say "I saw a big brown spider," rather than "I saw a brown big spider". However, the second sentence has the same literal meaning as the first.

Are there instances in which adjectives are non-commutative, so that permuting the order of the adjectives changes the literal meaning of what's being said?

It feels like there's a difference between adjectives as filtering devices "out of my college years, I mean the first of the horrible ones" and descriptive devices "this particular spider (which I have already referenced) was big and also brown". Of course you're right that filtering this way is non-commutative - and that answers the question as phrased in the title, but I don't think that's the distinction which applies to the example in the question body.
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GarethMar 20 '11 at 22:55

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These are called "restrictive" and "descriptive" adjectives, and generally English (unlike Spanish) doesn't make a distinction between them.
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Peter Shor Mar 22 '11 at 14:00

As pointed out elsewhere, “'local crumbed scallops' would be already-crumbed scallops obtained locally; 'crumbed local scallops' would be scallops obtained locally and then crumbed after being obtained”.

Generally speaking, there is a natural adjective order in English, which is everything but commutative. And “I saw a brown big spider” strikes me as ungrammatical.

Sure, as others have pointed out already, you can make it grammatical by putting the stress on brown, and regarding big spider as a single fixed unit. But that's precisely the thing: you have to jump through hoops, and even then it means something slightly different. (Yes, as you have pointed out in a comment, in a row of spiders it might still point out the exact same spider; but it is not saying the exact same thing about that spider.)

First off, an interesting aside. Someone else pondering your question led indirectly to the creation of the entire fantasy genre:

"I first tried to write a story when I was about seven. It was about a dragon. I remember nothing about it except a philological fact. My mother said nothing about the dragon, but pointed out that one could not say 'a green great dragon', but had to say 'a great green dragon'. I wondered why, and still do. The fact that I remember this is possibly significant, as I do not think I ever tried to write a story again for many years, and was taken up with language."

From "Letters of J.R.R. Tolkien".

To answer your question:

Are there instances in which adjectives are non-commutative, so that permuting the order of the adjectives changes the literal meaning of what's being said?

Yes, lots. For example, consider the mathematician's definition of "diameter". Suppose you have a shape -- any shape, in any number of dimensions -- and you want to find its diameter. How can we define diameter on an arbitrary shape?

It's actually straightforward. Consider every possible pair of points on the object, and consider every possible "path" between those two points. For each pair of points, find the shortest of those paths. The length of that path is the shortest distance between those two points. Now find the pair of points that has the longest shortest path between them. The length of that path is the diameter.

Clearly the "longest shortest path" is very different than the "shortest longest path".

See also my article about finding the longest shortest shortest path between poppyseeds on a bagel, if this subject interests you.

"Green great dragon" makes perfect sense if you want to distinguish this dragon from "green lesser dragons." Should dragons (or anything else, for that matter) be classified by color first or by size?
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oosterwalMar 21 '11 at 18:46

This may not be as drastic a change as you were hoping for, but it's at least fairly natural: sometimes the same word can serve either to describe one's opinion of the noun in question (which would put the adjective first in the order), or its purpose (which puts it last). Thus moving the adjective can change from one sense to the other - for example,

A red ornamental vase

is a vase whose purpose is to be ornamental (whether one's opinion is that it is decorative or not), while

An ornamental red vase

says that the vase is, in fact, aesthetically pleasing, whatever its original purpose (which may be merely holding plants).

Let's say you have some goats. Some of them are big goats, and a minority the big goats are prickly goats -- they have wiry hair that pricks you if you touch them. There is only one small goat, Napoleon, who is prickly. Some of the smaller goats are neither prickly nor soft, but generally the small goats are soft-haired. You might see a goat that is big for one of the small goats, and this is a big small goat. This is different from a small big goat, which is smaller than the biggest big goats, but still obviously a big goat. It might be a prickly big goat, or a small prickly goat. It might be a small prickly goat, and a softer one of the big prickly goats, but still obviously a prickly goat, and a big goat. Even if it is a small prickly big goat, it's still a big goat that is prickly. A prickly small goat might not be the same as a small prickly goat. If the small prickly goat is not Napoleon, then probably it is just a smaller big goat, i.e., a small prickly goat. That, or someone has described a small goat as having a rebarbative personality, though not literally a wiry coat.

+1 my head hurts. (I think the very first small big goat should be a big small goat?)
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psmearsMar 20 '11 at 7:41

@psmears -- oy! indeed. thanks. I think I broke that when I fixed something else (i.e., this may have been the result of a broken fix). Now I need to fix the broken fix (making it a fixed fix).
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jbelacquaMar 20 '11 at 7:47

Q1: What colour are spiders S1 and S4? Ans: Brown Q2. What size are spiders S1 and S4? Ans: Big. Ergo the spiders are both big and brown. Moreover, if S1 is brown, why isn't it in container B where it belongs?
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Mari-Lou AJan 7 '14 at 20:16

I agree, now that you point it out. The emphasis is quite different. I'm still curious about the possibility of a more drastic change in meaning. For example, if there were a lineup of spiders with different colors, some big and some small, I think the "big brown spider" and the "brown big spider" would probably be the same one. I'm wondering if there's a situation that's different. One example I thought of just now is that a "solitary blue Smurf" is literally blue, but a "blue solitary Smurf" is sad. That's the first interpretation that comes to mind for me, at least.
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Mark EichenlaubMar 20 '11 at 6:19

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How about a German East-Indian restaurant and an East-German Indian restaurant? One would be a German restaurant in India, and the other would be an Indian restaurant in East Germany (and given East Germany, that could mean North American Aboriginal OR East Indian...)
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migoMar 20 '11 at 14:33

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I realise I screwed that one up, but it could still work between East-German Indian restaurant and Indian East-German restaurant and East-Indian German restaurant and German East-Indian restaurant. One country adjective determines the nation, and the other the ethnicity.
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migoMar 20 '11 at 14:39